A microscopic analysis of shear acceleration

A microscopic analysis of the viscous energy gain of energetic particles in (gradual) non-relativistic shear flows is presented. We extend previous work and derive the Fokker-Planck coefficients for the average rate of momentum change and dispersion in the general case of a momentum-dependent scattering time $\tau(p) \propto p^{\alpha}$ with $\alpha \geq 0$. We show that in contrast to diffusive shock acceleration the characteristic shear acceleration timescale depends inversely on the particle mean free path which makes the mechanism particularly attractive for high energy seed particles. Based on an analysis of the associated Fokker-Planck equation we show that above the injection momentum $p_0$ power-law differential particle number density spectra $n(p) \propto p^{-(1+ \alpha)}$ are generated for $\alpha >0$ if radiative energy losses are negligible. We discuss the modifications introduced by synchrotron losses and determine the contribution of the accelerated particles to the viscosity of the background flow. Possible implications for the plasma composition in mildly relativistic extragalactic jet sources (WATs) are addressed.